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Let $P$ be a simple polygon with $n$ vertices, and let $q \in P$ be a pointin $P$. Let $k \in \{0, \dots, n - 1\}$. A point $p \in P$ is $k$-visible from$q$ if and only if the line segment $pq$ crosses the boundary of $P$ at most$k$ times. The $k$-visibility region of $q$ in $P$ is the set of all pointsthat are $k$-visible from $q$. We study the problem of computing the$k$-visibility region in the limited workspace model, where the input residesin a random-access read-only memory of $O(n)$ words, each with$\Omega(\log{n})$ bits. The algorithm can read and write $O(s)$ additionalwords of workspace, where $s \in \mathbb{N}$ is a parameter of the model. Theoutput is written to a write-only stream.  Given a simple polygon $P$ with $n$ vertices and a point $q \in P$, wepresent an algorithm that reports the $k$-visibility region of $q$ in $P$ in$O(cn/s+c\log{s} + \min\{\lceil k/s \rceil n,n \log{\log_s{n}}\})$ expectedtime using $O(s)$ words of workspace. Here, $c \in \{1, \dots, n\}$ is thenumber of critical vertices of $P$ for $q$ where the $k$-visibility region of$q$ may change. We generalize this result for polygons with holes and for setsof non-crossing line segments.

A Time-Space Trade-off for Computing the k-Visibility Region of a Point in a Polygon